Laurent Polynomials and Superintegrable Maps

Abstract
This article is dedicated to the memory of Vadim Kuznetsov,
and begins
with some of the author's recollections of him.
Thereafter, a brief review of Somos sequences is provided,
with particular focus
being made on the integrable structure of Somos-4 recurrences, and on
the Laurent property. Subsequently a family of fourth-order recurrences
that share the Laurent property are considered, which are equivalent to
Poisson maps in four dimensions. Two of these maps turn out
to be superintegrable, and their iteration furnishes
infinitely many solutions of some associated quartic Diophantine equations.